6 problems

Let $\mathbf{M}$ be the modular torus, the hyperbolic once-punctured torus described in the paper. A pair of simple closed geodesics is a pair of simple geodesics on $\mathbf{M}$ w…

Baragar–Bourgain–Gamburd–Sarnak conjecture. The Markoff equation $\mathbb{X}$ satisfies strong approximation at every prime $p$. Equivalently, $\operatorname{Out}(\Pi)$ acts transi…

Let $p$ be a prime, and let $\mathcal{G}_p$ be the Markoff graph over $\mathbb{F}_p$: its vertices are the nonzero triples in $\mathbb{F}_p^3$ satisfying … and two vertices are joi…

Let $\mathcal{M}$ be the set of Markoff triples over $\mathbb{Z}$, let $\mathcal{M}_p$ be the set of their reductions modulo a prime $p$, and let $\mathcal{G}^\times_p$ be the Mark…

Let $G_M$ be the graph whose vertices are the positive Markoff numbers, with edges arising from the Markoff transformations between coordinates of solutions to $x^2+y^2+z^2=3xyz$.…

Let $\mathcal M$ be the set of positive integer solutions to … and let $\mathcal M_p$ be the set of non-zero solutions modulo a prime $p$. The functional graph is generated by the…